{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:G6CXSWKLDKDAM55UM6WODWOVRH","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f7f730d978b38a9338069881b02f46ca9438aed84ead2c3f3a7ff099df1b23ea","cross_cats_sorted":["math.CO","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2011-12-14T18:02:03Z","title_canon_sha256":"1c14c9837f6ccefc9cb71df3bffb0c0a0b86b6e0025b7e05776a63dbda009820"},"schema_version":"1.0","source":{"id":"1112.3292","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.3292","created_at":"2026-05-18T03:46:18Z"},{"alias_kind":"arxiv_version","alias_value":"1112.3292v2","created_at":"2026-05-18T03:46:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.3292","created_at":"2026-05-18T03:46:18Z"},{"alias_kind":"pith_short_12","alias_value":"G6CXSWKLDKDA","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"G6CXSWKLDKDAM55U","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"G6CXSWKL","created_at":"2026-05-18T12:26:28Z"}],"graph_snapshots":[{"event_id":"sha256:66fc9965f02516d821ab7677806331a2c90dad04fd145f37b31893050bb971c5","target":"graph","created_at":"2026-05-18T03:46:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We prove that for a finitely generated infinite nilpotent group G with a first order structure (G,*,...), the connected component G*0 of a sufficiently saturated extension G* of G exists and equals $\\bigcap_{n\\in\\N} {g^n : g\\in G^*}$. We construct a first order expansion of Z by a predicate (Z,+,P) such that the type-connected component Z*00_{\\emptyset} is strictly smaller than Z*0. We generalize this to finitely generated virtually solvable groups. As a corollary of our construction we obtain an optimality result for the van der Waerden theorem.","authors_text":"Cong Chen, Jakub Gismatullin, Nathan Bowler","cross_cats":["math.CO","math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2011-12-14T18:02:03Z","title":"Model theoretic connected components of finitely generated nilpotent groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.3292","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:87db8c62d8aadfe962366a887c4ab69639ee4dc7e25144a345029a262cfdab00","target":"record","created_at":"2026-05-18T03:46:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f7f730d978b38a9338069881b02f46ca9438aed84ead2c3f3a7ff099df1b23ea","cross_cats_sorted":["math.CO","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2011-12-14T18:02:03Z","title_canon_sha256":"1c14c9837f6ccefc9cb71df3bffb0c0a0b86b6e0025b7e05776a63dbda009820"},"schema_version":"1.0","source":{"id":"1112.3292","kind":"arxiv","version":2}},"canonical_sha256":"378579594b1a860677b467ace1d9d589e4c4036e1c939474b48880a7dbdac93b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"378579594b1a860677b467ace1d9d589e4c4036e1c939474b48880a7dbdac93b","first_computed_at":"2026-05-18T03:46:18.492382Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:46:18.492382Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fGToVE0S0lIa9Vamd/OrZ5a77UdkCSN0sRt+w4z4Qa+JEPqUs6Pxe9hMNus1bJJwAybiPdVr2U2TFxP87MD/Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:46:18.493064Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.3292","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:87db8c62d8aadfe962366a887c4ab69639ee4dc7e25144a345029a262cfdab00","sha256:66fc9965f02516d821ab7677806331a2c90dad04fd145f37b31893050bb971c5"],"state_sha256":"62754e551d419d097bdb26bec4056970a0356ea9b00fffd6ac61b586922a417b"}